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Math Help - Discrete Math, Please Help!!

  1. #1
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    Discrete Math, Please Help!!

    Prove the sets A and B defined below are equal.

    A = {(a, b) is a member of Z+ Z+|a is a factor of b}

    B is defined recursively by

    Initial Step: (1, 1) is a member of B
    Recursive Step: If (a, b) is a member of B and c is a member of Z+ then
    (ac, bc) is a member of B and (a, bc) is a member of B
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  2. #2
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    Quote Originally Posted by jas05s View Post
    Prove the sets A and B defined below are equal.
    A = {(a, b) is a member of Z+ Z+|a is a factor of b}
    B is defined recursively by
    Initial Step: (1, 1) is a member of B
    Recursive Step: If (a, b) is a member of B and c is a member of Z+ then
    (ac, bc) is a member of B and (a, bc) is a member of B
    We know that (x,y) \in A \Rightarrow \quad \left( {\exists j \in Z^ +  } \right)\left[ {xj = y} \right].
    So by definition we get (1,1) \in B \Rightarrow \quad (x,x) \in B,\,\& \,(x,jx) \in B.
    This means that (x,y) \in B\mbox{ so } A \subseteq B.

    It is simple now to prove B \subseteq A.
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